Mesmerizing. It's really interesting to see how design engineers create processes for manufacturing.

#1 Yes that liquid is called "coolant" and is used when machining or in areas where there's friction/heat. If you produce too much heat into the material during the process it'll change the condition (or properties) of the material often in a bad way whereby it becomes weakened. The less "damage" you create during the making of the widget, the less post operations you need to return it to the condition required for it's intended application/use.

It would also act as a lubricant. In the case of the lathe with all of those tools it was indexing 2:14 , if you don't keep the area cool and lubed, the tool bits would heat up and dull during the machining requiring replacing those "bits" with new ones shockingly often. At home, if you're drilling something like steel, if you use a little bit of oil or cutting fluid during the process the drilling will go easier and your drill bit will stay sharper longer.

The coolant is water and a prescribed amount of corrosion inhibitor protecting the material (if applicable) and the machinery from rust. If you just used straight water, the machinery would rust very quickly and obviously the machinery would be very expensive to replace.